Casimir-mediated amplification of flexural modes in suspended graphene.

We develop a minimal theoretical framework to describe near-field Casimir-mediated coupling between two suspended nanomembranes: a graphene sheet and a gold-coated GaAs membrane separated by a micrometer-scale vacuum gap. Coherent excitation of flexural (out-of-plane) modes in the gold membrane, provided by an external piezoelectric drive, enables efficient energy transfer to graphene through static vacuum electromagnetic coupling. Using a coupled-mode description equivalent to the classical equations of motion, we identify hybridized flexural modes exhibiting avoided crossings, tunable spectral response, and drive-dependent energy exchange. Under experimentally accessible conditions, Casimir-mediated mode hybridisation transfers energy from the driven gold membrane to graphene at rates comparable to or exceeding the intrinsic mechanical losses of graphene, resulting in a strong enhancement of its flexural response and linewidth narrowing. The Casimir interaction acts here solely as a static, non-contact coupling mechanism, while all energy is supplied externally through coherent mechanical driving, yielding drive-induced amplification without intrinsic negative damping. Our results establish Casimir coupling as a practical resource for controlling flexural dynamics in graphene nanomembranes and provide a platform for non-contact energy transfer and mode-selective amplification in two-dimensional mechanical systems.